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We study a differential-difference equation of the form t x ͑n +1͒ = f͑t͑n͒ , t͑n +1͒ , t x ͑n͒͒ with unknown t = t͑n , x͒ depending on x and n. The equation is called a Darboux integrable if there exist functions F ͑called an x-integral͒ and I ͑called an n-integral͒, both of a finite number of variables x , t͑n͒ , t͑n Ϯ 1͒ , t͑n Ϯ 2͒ , ... , t x ͑n͒ , t xx ͑n͒ ,..., such that D x F = 0 and DI = I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp͑n͒ = p͑n +1͒. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f͑x , y , z͒ = z + d͑x , y͒.

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Cited by 2 publications

References 9 publications

On the characteristic Lie algebras for equations u xy = f(u, u x )

On the characteristic Lie algebras for equations u xy = f(u, u x )

On the classification of Darboux integrable chains

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